Book: Algorithms on strings, trees and sequences by Dan Gusfield Presented by: Amir Anter and Vladimir Zoubritsky

 Given a string P called pattern and a longer string T called the text, the exact matching problem is to find all occurrences, if any, of pattern P in text T.

 P=aa and T=abaabaaa  P occurs in T 3 times, starting at locations 3,6 and 7. ◦ Location 3:  abaabaaa ◦ Location 6:  abaabaaa ◦ Location 7:  abaabaaa  Please note that the occurrences may overlap, locations 6,7.

 Grep command in Unix: ◦ grep apple fruitlist.txt  Internet browsers – Find option.  Biology - Searching for a string in a DNA database.  Articles, online books.

1. Align the left end of P with the left end of T. 2. compares the characters of P and T left to right until: 2.1 A mismatch 2.2 P ends – An occurrence of P is reported. 3. P is shifted one place to the right. 4. If P’s right end is farther than T’s right end: Finish 5.Else Go to 2

Step 1: abaabaaa aa Step 1.1: abaabaaaa Step 1.2: abaabaaaa

Step 2: abaabaaa aa Step 2.1: abaabaaa aa

Step 3: abaabaaa aa Step 3.1: abaabaaa aa Step 3.2: abaabaaa aa Report match at location 3

Step 4: abaabaaa aa Step 4.1: abaabaaa aa Step 4.2: abaabaaa aa

Step 5: abaabaaa aa Step 5.1: abaabaaa aa

Step 6: abaabaaa aa Step 6.1: abaabaaa aa Step 6.2: abaabaaa aa Report match at location 6

Step 7: abaabaaa aa Step 7.1: abaabaaa aa Step 7.2: abaabaaa aa Report match at location 7

Step 8: abaabaaa aa End

 Let P’s length be n.  Let T’s length be m.  Number of character comparisons in the worst case is O(nm).  No additional storage is needed.  30 character string search in GenBank (DNA DB) took more than 4 hours.  We will shows a linear lime algorithm, which improves this time to 10 minutes.

 Given a string S and a position, let be the length of the longest substring of S that starts at i and matches a prefix of S.  Equivalently: is the length of the longest prefix of S[i..|S|] that matches a prefix of S.

aabcaabxaaz

P – pattern of length n. T – text of length m. Let S = P$T, where $ does not appear in P and in T. S’s length is. Lets assume we have computed for at a preprocessing stage. Claim: Any value of i>n+1 such that indentifies an occurrence of P in T starting at position i-(n+1) of T. Claim: If P occurs in T starting at position j of T, then Do we really need $? (Except for USD )

 For any position where, Z-box at i is defined as the interval starting at i and ending at a a b c a a b x a a z

- The right-most end of any Z-box that begins up to position i-1. - A substring - some Z-box ending at. - The left end of some. i S

a a b c a a b x a a z

Our task is to compute Z values in linear time. Let’s find by comparing left to right characters of and until a mismatch is found. is the length of the matching string.

Let’s assume we have all Z values to k-1. The idea is to use already computed Z values to compute.

xx Let’s assume

 Example - JavaScript Example - JavaScript



Case 1: K>r

Case 2.a

Case 2.b

Do we really need $ ? By definition, is the length of the longest prefix of S[i..|S|] that matches a prefix of S. If P length is n, indicates an occurrence of P in T, in case S=P$T and also in case S=PT. The answer is no, it terms of correctness.

So why to use $? Using $ ensures a limit of n for the values of In The algorithm we use some and to compute the current. We need only additional space. is not bearable.

 iterations  Number of compressions: ◦ Each mismatch ends an iteration. Max total of mismatches for the entire algorithm. ◦ Each match increments the value of r at least by 1. ◦ number of matches comparisons for the entire algorithm

 Knuth-Morris-Pratt (KMP)  Aho–Corasick string matching algorithm ◦ Is a generalization of KMP. ◦ Set of patterns in linear time.  Boyer-Moore ◦ Typically runs in sublinear time. ◦ It is used in practice for exact matching. ◦ Worst case linear.

Thank You!